Consider the set of homeomorphisms of the topological n-ball to itself with the compact open topology. Sitting inside this space of homeomorphisms are particular subspaces. The first subspace is those homeomorphisms such that the restriction to the boundary is the identity. A path in this space is the same as an isotopy. Alexander's trick tells us that this space is path connected. In other words, if we are given any two homeomorphisms that restrict to the identity on the boundary, then these two homeomorphisms are isotopic relative to the boundary. If we push Alexander's trick a bit further, we can show that if we have two homeomorphisms of the n-ball such that the homeomorphisms restricted to the boundary are isotopic, then we may extend to an isotopy of the two homeomorphisms on the entire n-ball. My questions are as follows:
- What is the homotopy type of the space of homeomorphisms of the n-ball such that the homeomorphism restricted to the boundary is the identity? In particular, is this space contractible?
- Same question for the space of homeomorphisms such that the homeomorphisms restricted to the boundary is isotopic to the identity (on the boundary). In other words, what is the homotopy type of the space of homeomorphisms such that the homeomorphisms restricted to the boundary is isotopic to the identity? In particular is this space contractible?
- What is a good reference for all of this (including, but not limited to Alexander's trick).
As for Alexander's trick, the Wikipedia page is fairly good, but not a proper source.